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An Analysis of Phase Noise and Fokker-Planck Equations. Hao-Min Zhou School of Mathematics Georgia Institute of Technology. Joint work with Shui-Nee Chow. International conference of random dynamical systems, Tianjin, China, June 8-12, 2009. Partially Supported by NSF. Outline. - PowerPoint PPT Presentation
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An Analysis of Phase Noise and Fokker-Planck Equations
Hao-Min ZhouSchool of Mathematics
Georgia Institute of Technology
Partially Supported by NSF
Joint work with Shui-Nee Chow
International conference of random dynamical systems, Tianjin, China, June 8-12, 2009
Outline
• Introduction and motivation
• Moving coordinate transforms
• Phase noise equations and Fokker-Planck equations
• Example: van der Pol oscillators and ACD
• Conclusion
Introduction and Motivation
• A orbital stable periodic solution (limit cycle) (with period ) of a differential system
• Phase noise is caused by perturbations, which are unavoidable in practice: the solution doesn’t return to the starting point after a period .
• Phase noise usually persists, may become large.
• Phase noise is important in many areas including circuit design,
and optics.
),(
),(
212
211
uugdt
du
uufdt
duT
Oscillators
• Phase noise in nonlinear electric oscillators:
• Small noise can lead to dramatic spectral changes
• Many undesired problems associated with phase noise, such as interchannel interference and jitter.
Analog to Digital Converter (ADC)
5 7
• ADC is essential for wireless communications.• Input: wave (amplitude, frequency). Output: digit computed in real-time, during one single period (number of spikes).• Effect of the noise in the transmission system.
correct output wrong output
5 7 5 8
Bit Error Rate (BER) : ratio of received bits that are in error, relative to the amount of bits received. BER expressed in log scale (dB).
ADC Example
)],(-[1
'
]-)(['
ygxy
ytfx
,
0,
0,
)(
iyy
yiKy
yy
yg
A piecewise linear ADC model is
The input is an analog signal, i.e. ttf sin)(
The output is the number of spikes in a period, which realizes the conversion of analog signals to digital ones.
Our goals
• Establish a framework to rigorously analyze phase noise from both dynamic system and probability perspectives.
• Develop numerical schemes to compute phase noise, which are useful tools for system design.
• Estimate Shannon entropy curves to evaluate the performance of practical systems
Approaches
)()()(
)()()(
22
11
twtuty
twtutx
))(),(( tytx
))(),(( 21 tutu
• Traditional nonlinear analysis based on linearization is invalid: decompose the perturbed solution
where is the unperturbed solution and is the deviation, then the error satisfies
))(),(( 21 twtw
)())(()()( tbtuBtwtAdt
dw
• The deviation can grow to infinitely large (even amplitude error remains small for stable systems, but phase error can be large)
)(tw
•The system is self-sustained, and must have one as its eigenvalue.
)(tA
Approaches
• A conjecture: decompose perturbations into two (orthogonal) components, one along the tangent, one along normal direction, perturbations along tangent generates purely phase noise and normal component causes only amplitude deviation, Hajimiri-Lee (’97).
•This conjecture is not valid, Demir-Roychowdhury (’98). Perturbation orthogonal to the orbit can also cause phase deviation.
Approaches• Large literature is available for individual systems, such as pumped lasers by Lax (’67), but lack of general theory for phase noise. • Two appealing approaches:
1. Model the perturbed systems by SDE’s and derive the associated Fokker-Planck equations, then use asymptotic analysis to estimate the leading contributions of transition probability distribution function , i.e. in Limketkai (’05), the leading term is approximated by a gaussian:
2
),(),,( ExCex
where satisfy a diffusion PDE),( ,0 BA
and are coefficients obtained in asymptotic expansions
ECBA ,,,
Approaches
)())(()( tyttutx
2. Decompose oscillator response into phase and magnitude components and obtain equations for the phase error, for examples: Kartner (’90), Hajimiri-Lee (’98),Demir-Mehrotra-Roychowdhury (’00), i.e.
where is defined by a SDE depending on the largest eigenvalue and eigenfunction of state transition matrix in Floquet theory:
)(t
1
( )( ( )) ( ( ( )))T
t
d tv t t B u t t dW
dt
1 1v
may grow to infinitely large even for small perturbations)(t
Moving Orthogonal Systems
),,(),(
),,(),(
tyxkyxgdt
dy
tyxhyxfdt
dx
• A moving orthogonal coordinate systems along
• Consider solutions of the perturbed systems
))(),(( tytx
are small perturbations),,(),,,( tyxktyxh
Equations for the new variables
)())(())((
))((
)(
)(
2
1 ttztu
tu
ty
tx
)(
)(
)(
)(:
ty
tx
t
t
• Solutions of the perturbed system can be represented by
denoted by
• For small perturbations, this transform is invertible and both forward and inverse transforms are smooth.
• Two components and are not orthogonal, which is different from the usual orthogonal decompositions.
))(( tu ))(( tz
Equations for the new variables
)(t
))()((1
))()((
kgfhfgrdt
d
kgghffr
s
dt
d
• The new phase and amplitude deviation satisfy (Hale (’67)))(t
where notations are
),( 222 gfr 2
''
r
gffgw
,)( 1 wrs
)),,(),,((
)),,(),,((
)),(),,((
)),(),,((
tyxkk
tyxhh
yxgg
yxff
,))(),((
))(),((
21
21
uugg
uuff
Evaluate on the unperturbed orbit
Evaluate on the perturbed orbit
Stochastic Perturbations
22
112
22
111
tt
tt
dWdWdtd
dWdWdtd
• Perturbations in oscillators are random, which are often modeled by
Where are independent Brownian motions.
2
1
),()),,(),((
),()),,(),((
t
t
dWYXbdttYXkYXgdY
dWYXadttYXhYXfdX
21, tt WW
• The transform becomes),())((
))((
))((
)(
)(
)(
)(
2
1 ttztu
tu
t
t
tY
tX
• Theorem 1: if stay close to , then remain as Ito processes and satisfy
)(),( tYtX )(),( tt
Stochastic Perturbations
• Theorem 2: the transition probability of satisfies the Fokker-Planck equation
)(),( tt ),,( tp
• The coefficients are
)))(())((2))(((2
1)()( 2
2212211
22
2121 ppppppt
with initial condition)()( 00
t
p
bfr
agr
bgr
s
afr
s
1
1
2
1
2
1
,))()((
2)()((
1
))(2
))()((2
)()((
222
22221
bgafr
wskgfhfg
r
abr
wsfgbgaf
r
skgghff
r
s
Stochastic Perturbations
• Theorem 4: the transition probability of satisfies the Fokker-Planck equation
)(),( tt
),,( tp
• For a general problem in
)),((2
1)( ppp T
t
where
,2
1
)())(())(()( ttztutX
tWdXadttXhXfXd
)()),()((
nR
The solution can also be transformed into where
)1( nnRz
,2
1
t
t
Wddtd
Wddtd
• Theorem 3: if stay close to , then remain as Ito processes and satisfy
)(tX
)(),( tt
where can be determined similarly.nnnn RRR )1(121 ,,,
van der Pol Oscillators
vvqv
vq
)1( 2
qv
• Unperturbed van der Pol Oscillators are often described by
introduce new variable
0)1( 2 qqqq
the equation becomes
• In practice, noise enters the system, which is model by
by introducing the new variable , the system becomes
tdWYYXdY
YdX
)1( 2
• Both and are positive small constant numbers, it is interesting to study the case eventually.
dXY
0)1( 2 tdWqqqq
van der Pol OscillatorsAssume are small (in oscillators, the periodic orbits are stable, and perturbations of amplitude will remain small, i.e. is small). The leading term system is
The corresponding Fokker-Planck equation is
By the method of averaging for stochastic equations, it is equivalent to
t
t
dWdtd
dWdtd
sin)sin)sin41((
cos)3
4
3
2(
22
t
t
dWdtd
dWdtd
sin
cos)3
4
3
2(
)))sin(())2sin()4
3
2
3(())cos)
4
3
2
3(((2
1)( 222222
ppppppt
van der Pol Oscillators
1. Impuse noise in current at the peak of current (zero voltage),
2. Impose noise in current at the peak of voltage (zero current),
Two interesting observations (made by engineers, Hajimiri-Lee(’98), Limketkai(’05 ) ):
,0 , ,0sin
Perturbation has no impact on amplitude, and maximum impact on phase noise.
,1cos
,2
,
2
3 ,0cos
Noise has no impact on phase, and maximum impact on amplitude error.
,1sin
van der Pol Oscillators
ttdWdtd sin
The dynamic of amplitude error can be approximated by
which leads to the following properties if the initial is small:
• The mean: .• The variance:
• It is a Gaussian variable.
0)( E
2
21
2222
2
0
)(222
))2sin1
2cos2
1()
1
2
1(
2
1)1((
sin)(
ttee
sdseE
tt
tst
as t
tsYststs
log)(sup)(sup 2
00
tdWdttYtdY )()(
This implies that if , then for any given 2
et
)(sup0
sts
.
The amplitude error also satisfies:
where
Conclusion
• A general framework, based on a moving orthogonal coordinate system, has been established to rigorously study the phase and amplitude noise.
• Both dynamic equations and Fokker-Planck equations for the phase noise are derived.
• The general theory has been applied to the van der Pol oscillators. Derived equations can explain some interesting observations in practice.
